# Properties

 Label 3040.2.a.p Level $3040$ Weight $2$ Character orbit 3040.a Self dual yes Analytic conductor $24.275$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3040,2,Mod(1,3040)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3040, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("3040.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$3040 = 2^{5} \cdot 5 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3040.a (trivial)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: yes Analytic conductor: $$24.2745222145$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.148.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - x^{2} - 3x + 1$$ x^3 - x^2 - 3*x + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta_1 + 1) q^{3} + q^{5} + ( - \beta_{2} + \beta_1 - 1) q^{7} + (\beta_{2} + 3 \beta_1) q^{9}+O(q^{10})$$ q + (b1 + 1) * q^3 + q^5 + (-b2 + b1 - 1) * q^7 + (b2 + 3*b1) * q^9 $$q + (\beta_1 + 1) q^{3} + q^{5} + ( - \beta_{2} + \beta_1 - 1) q^{7} + (\beta_{2} + 3 \beta_1) q^{9} + (\beta_{2} + \beta_1 + 3) q^{11} + ( - \beta_{2} + 2 \beta_1 - 2) q^{13} + (\beta_1 + 1) q^{15} + 2 q^{17} - q^{19} + 2 q^{21} + (\beta_{2} - \beta_1 + 1) q^{23} + q^{25} + (4 \beta_{2} + 4 \beta_1 + 2) q^{27} + ( - 4 \beta_{2} - 2 \beta_1 - 4) q^{29} + ( - 2 \beta_{2} - 4 \beta_1 + 2) q^{31} + (2 \beta_{2} + 6 \beta_1 + 4) q^{33} + ( - \beta_{2} + \beta_1 - 1) q^{35} + ( - \beta_{2} + 2 \beta_1) q^{37} + (\beta_{2} + \beta_1 + 3) q^{39} + ( - 4 \beta_{2} - 2 \beta_1 + 2) q^{41} + ( - 3 \beta_{2} - 7 \beta_1 + 7) q^{43} + (\beta_{2} + 3 \beta_1) q^{45} + ( - \beta_{2} + \beta_1 - 1) q^{47} + (2 \beta_{2} - 4 \beta_1 + 1) q^{49} + (2 \beta_1 + 2) q^{51} + (\beta_{2} - 2 \beta_1 - 4) q^{53} + (\beta_{2} + \beta_1 + 3) q^{55} + ( - \beta_1 - 1) q^{57} + ( - 2 \beta_{2} - 2 \beta_1 + 4) q^{59} + (7 \beta_{2} + \beta_1 + 3) q^{61} + (3 \beta_{2} - \beta_1 + 5) q^{63} + ( - \beta_{2} + 2 \beta_1 - 2) q^{65} + ( - 4 \beta_{2} - 3 \beta_1 - 1) q^{67} - 2 q^{69} + ( - 2 \beta_{2} - 4 \beta_1 + 2) q^{71} + ( - 4 \beta_{2} - 2 \beta_1 + 4) q^{73} + (\beta_1 + 1) q^{75} + ( - 2 \beta_{2} + 4 \beta_1 - 4) q^{77} + (6 \beta_{2} + 2 \beta_1 + 4) q^{79} + (5 \beta_{2} + 5 \beta_1 + 6) q^{81} + (3 \beta_{2} - 5 \beta_1 + 1) q^{83} + 2 q^{85} + ( - 6 \beta_{2} - 12 \beta_1 - 4) q^{87} + (4 \beta_{2} + 2 \beta_1) q^{89} + (4 \beta_{2} - 6 \beta_1 + 12) q^{91} + ( - 6 \beta_{2} - 8 \beta_1 - 4) q^{93} - q^{95} + (3 \beta_{2} + 4 \beta_1 + 2) q^{97} + (5 \beta_{2} + 15 \beta_1 + 5) q^{99}+O(q^{100})$$ q + (b1 + 1) * q^3 + q^5 + (-b2 + b1 - 1) * q^7 + (b2 + 3*b1) * q^9 + (b2 + b1 + 3) * q^11 + (-b2 + 2*b1 - 2) * q^13 + (b1 + 1) * q^15 + 2 * q^17 - q^19 + 2 * q^21 + (b2 - b1 + 1) * q^23 + q^25 + (4*b2 + 4*b1 + 2) * q^27 + (-4*b2 - 2*b1 - 4) * q^29 + (-2*b2 - 4*b1 + 2) * q^31 + (2*b2 + 6*b1 + 4) * q^33 + (-b2 + b1 - 1) * q^35 + (-b2 + 2*b1) * q^37 + (b2 + b1 + 3) * q^39 + (-4*b2 - 2*b1 + 2) * q^41 + (-3*b2 - 7*b1 + 7) * q^43 + (b2 + 3*b1) * q^45 + (-b2 + b1 - 1) * q^47 + (2*b2 - 4*b1 + 1) * q^49 + (2*b1 + 2) * q^51 + (b2 - 2*b1 - 4) * q^53 + (b2 + b1 + 3) * q^55 + (-b1 - 1) * q^57 + (-2*b2 - 2*b1 + 4) * q^59 + (7*b2 + b1 + 3) * q^61 + (3*b2 - b1 + 5) * q^63 + (-b2 + 2*b1 - 2) * q^65 + (-4*b2 - 3*b1 - 1) * q^67 - 2 * q^69 + (-2*b2 - 4*b1 + 2) * q^71 + (-4*b2 - 2*b1 + 4) * q^73 + (b1 + 1) * q^75 + (-2*b2 + 4*b1 - 4) * q^77 + (6*b2 + 2*b1 + 4) * q^79 + (5*b2 + 5*b1 + 6) * q^81 + (3*b2 - 5*b1 + 1) * q^83 + 2 * q^85 + (-6*b2 - 12*b1 - 4) * q^87 + (4*b2 + 2*b1) * q^89 + (4*b2 - 6*b1 + 12) * q^91 + (-6*b2 - 8*b1 - 4) * q^93 - q^95 + (3*b2 + 4*b1 + 2) * q^97 + (5*b2 + 15*b1 + 5) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + 4 q^{3} + 3 q^{5} - 2 q^{7} + 3 q^{9}+O(q^{10})$$ 3 * q + 4 * q^3 + 3 * q^5 - 2 * q^7 + 3 * q^9 $$3 q + 4 q^{3} + 3 q^{5} - 2 q^{7} + 3 q^{9} + 10 q^{11} - 4 q^{13} + 4 q^{15} + 6 q^{17} - 3 q^{19} + 6 q^{21} + 2 q^{23} + 3 q^{25} + 10 q^{27} - 14 q^{29} + 2 q^{31} + 18 q^{33} - 2 q^{35} + 2 q^{37} + 10 q^{39} + 4 q^{41} + 14 q^{43} + 3 q^{45} - 2 q^{47} - q^{49} + 8 q^{51} - 14 q^{53} + 10 q^{55} - 4 q^{57} + 10 q^{59} + 10 q^{61} + 14 q^{63} - 4 q^{65} - 6 q^{67} - 6 q^{69} + 2 q^{71} + 10 q^{73} + 4 q^{75} - 8 q^{77} + 14 q^{79} + 23 q^{81} - 2 q^{83} + 6 q^{85} - 24 q^{87} + 2 q^{89} + 30 q^{91} - 20 q^{93} - 3 q^{95} + 10 q^{97} + 30 q^{99}+O(q^{100})$$ 3 * q + 4 * q^3 + 3 * q^5 - 2 * q^7 + 3 * q^9 + 10 * q^11 - 4 * q^13 + 4 * q^15 + 6 * q^17 - 3 * q^19 + 6 * q^21 + 2 * q^23 + 3 * q^25 + 10 * q^27 - 14 * q^29 + 2 * q^31 + 18 * q^33 - 2 * q^35 + 2 * q^37 + 10 * q^39 + 4 * q^41 + 14 * q^43 + 3 * q^45 - 2 * q^47 - q^49 + 8 * q^51 - 14 * q^53 + 10 * q^55 - 4 * q^57 + 10 * q^59 + 10 * q^61 + 14 * q^63 - 4 * q^65 - 6 * q^67 - 6 * q^69 + 2 * q^71 + 10 * q^73 + 4 * q^75 - 8 * q^77 + 14 * q^79 + 23 * q^81 - 2 * q^83 + 6 * q^85 - 24 * q^87 + 2 * q^89 + 30 * q^91 - 20 * q^93 - 3 * q^95 + 10 * q^97 + 30 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 3x + 1$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - \nu - 2$$ v^2 - v - 2
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + \beta _1 + 2$$ b2 + b1 + 2

## Embeddings

For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 −1.48119 0.311108 2.17009
0 −0.481194 0 1.00000 0 −4.15633 0 −2.76845 0
1.2 0 1.31111 0 1.00000 0 1.52543 0 −1.28100 0
1.3 0 3.17009 0 1.00000 0 0.630898 0 7.04945 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$1$$
$$5$$ $$-1$$
$$19$$ $$1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3040.2.a.p yes 3
4.b odd 2 1 3040.2.a.j 3
8.b even 2 1 6080.2.a.bm 3
8.d odd 2 1 6080.2.a.ca 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3040.2.a.j 3 4.b odd 2 1
3040.2.a.p yes 3 1.a even 1 1 trivial
6080.2.a.bm 3 8.b even 2 1
6080.2.a.ca 3 8.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(3040))$$:

 $$T_{3}^{3} - 4T_{3}^{2} + 2T_{3} + 2$$ T3^3 - 4*T3^2 + 2*T3 + 2 $$T_{7}^{3} + 2T_{7}^{2} - 8T_{7} + 4$$ T7^3 + 2*T7^2 - 8*T7 + 4 $$T_{11}^{3} - 10T_{11}^{2} + 28T_{11} - 20$$ T11^3 - 10*T11^2 + 28*T11 - 20

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3}$$
$3$ $$T^{3} - 4 T^{2} + \cdots + 2$$
$5$ $$(T - 1)^{3}$$
$7$ $$T^{3} + 2 T^{2} + \cdots + 4$$
$11$ $$T^{3} - 10 T^{2} + \cdots - 20$$
$13$ $$T^{3} + 4 T^{2} + \cdots + 10$$
$17$ $$(T - 2)^{3}$$
$19$ $$(T + 1)^{3}$$
$23$ $$T^{3} - 2 T^{2} + \cdots - 4$$
$29$ $$T^{3} + 14 T^{2} + \cdots - 344$$
$31$ $$T^{3} - 2 T^{2} + \cdots + 184$$
$37$ $$T^{3} - 2 T^{2} + \cdots + 50$$
$41$ $$T^{3} - 4 T^{2} + \cdots - 80$$
$43$ $$T^{3} - 14 T^{2} + \cdots + 1388$$
$47$ $$T^{3} + 2 T^{2} + \cdots + 4$$
$53$ $$T^{3} + 14 T^{2} + \cdots - 34$$
$59$ $$T^{3} - 10 T^{2} + \cdots + 40$$
$61$ $$T^{3} - 10 T^{2} + \cdots + 1444$$
$67$ $$T^{3} + 6 T^{2} + \cdots - 218$$
$71$ $$T^{3} - 2 T^{2} + \cdots + 184$$
$73$ $$T^{3} - 10 T^{2} + \cdots + 8$$
$79$ $$T^{3} - 14 T^{2} + \cdots + 1112$$
$83$ $$T^{3} + 2 T^{2} + \cdots - 796$$
$89$ $$T^{3} - 2 T^{2} + \cdots + 200$$
$97$ $$T^{3} - 10 T^{2} + \cdots + 46$$